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1、Logistic regressionLogistic回归回归Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 第一节第一节. .非条件非条件logisticlogistic回归回归第二节第二节. .条件条件logisticlogistic回归回归第三节第三节. . 应用及其注意事项应用及其注意事项Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, Augu
2、st 21, August 21, 2024202420242024 医学研究中常碰到应变量应变量的可能取值仅有两个（即二分类变量二分类变量），如发病与未发病、阳性与阴性、死亡与生存、治愈与未治愈、暴露与未暴露等，显然这类资料不满足多元（重）回归的条件 什么情况下采用什么情况下采用LogisticLogistic回归回归Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 Brown(1980)在术前检查了53例前列腺癌患者，拟用年龄(AG
3、E)、酸性磷酸酯酶(ACID)两个连续型的变量，X射线(X_RAY)、术前探针活检病理分级(GRADE)、直肠指检肿瘤的大小与位置(STAGE)三个分类变量与手术探查结果变量NODES（1、0分别表示癌症淋巴结转移与未转移 ）建立淋巴结转移的预报模型。实例Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024（一）53例接受手术的前列腺癌患者情况 Wednesday, Wednesday, Wednesday, Wednesday, Augu
4、st 21, August 21, August 21, August 21, 2024202420242024（二）26例冠心病病人和28例对照进行病例对照研究 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 202420242024202426例冠心病病人和28例对照者进行病例对照研究 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 20
5、24202420242024一、logistic回归模型 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024概率预报模型概率预报模型 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024二、模型的参数估计参数估计 Logistic回归参数的估计通常采用最大似然法最大似然法(maximu
6、m likelihood，ML)。最大似然法的基本思想是先建立似然函数与对数似然函数，再通过使对数似然函数最大求解相应的参数值，所得到的估计值称为参数的最大似然估计值。 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024参数估计的公式参数估计的公式 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 20242024
7、20242024三、回归三、回归参数的假设检验参数的假设检验 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024优势比及其可信区间优势比及其可信区间 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024标准化回归标准化回归参数参数用于评价各自变量对模型的贡献大小用于评价各自变量对模型的
8、贡献大小Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024SAS程序程序Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 The LOGISTIC ProcedureAnalysis of Maximum Likelihood EstimatesWednesday, Wednesd
9、ay, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 预报模型预报模型Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 The LOGISTIC ProcedureAnalysis of Maximum Likelihood EstimatesWednesday, Wednesday, Wednesday, Wedne
10、sday, August 21, August 21, August 21, August 21, 2024202420242024 预报模型预报模型Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024四、回归四、回归参数的意义参数的意义 当只有一个自变量时，以相应的预报概率 为纵轴，自变量 为横轴，可绘制出一条S形曲线。回归参数的正负符号与绝对值大小，分别决定了S形曲线的方向与形状Wednesday, Wednesday, Wednesd
11、ay, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024优势比改变优势比改变exp(exp(b bj j) )个单位个单位Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21,
12、2024202420242024Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024五、整个回归模型五、整个回归模型的假设检验的假设检验 Wednesday, Wednesday, Wednesday, Wednesday, Augus
13、t 21, August 21, August 21, August 21, 2024202420242024似然比检验（似然比检验（likelihood ratio test）Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024ROC曲线模型评价曲线模型评价Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024
14、202420242024ROC曲线模型评价曲线模型评价图图16-2 Logistic16-2 Logistic回归预报能力的回归预报能力的ROCROC曲线曲线Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024六、六、logistic逐步回归（变量筛选）逐步回归（变量筛选）MODEL语句加入选项“ SELECTION=STEPWISE SLE=0.100.10 SLS=0.100.10;”常采用似然比检验：决定自变量是否引入或剔除。Wedn
15、esday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024模型中有模型中有X5、X6、X8，看是否引入看是否引入X1模型含X5、X6、X8的模型的负二倍对数似然为： 50.402模型含X1、X5、X6、X8的模型的负二倍对数似然为： 46.224Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024第二
16、节第二节.条件条件logistic回归回归 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024条件似然函数条件似然函数 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 20242024202420241:3配对的例子配对的例子 Wednesday, Wednesday, Wednesday, Wednesday,
17、 August 21, August 21, August 21, August 21, 20242024202420241:2配对的例子配对的例子 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024表表16-7条件条件logistic回归的回归的SAS程序程序 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2
18、024202420242024结果结果Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 第三节第三节 应用及其注意事项应用及其注意事项应变量应变量为（二项）分类的资料为（二项）分类的资料（预测、判别、危险因素分析等等）（预测、判别、危险因素分析等等）Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202
19、420242024 注意事项注意事项1.分类自变量的哑变量编码 2. 为了便于解释，对二项分类变量一般按0、1编码，一般以0表示阴性或较轻情况，而1表示阳性或较严重情况。如果对二项分类变量按+1与-1编码，那么所得的 ，容易造成错误的解释。 Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024西、中西、中三种疗法哑变量化 原资料原资料姓名姓名性性别年年龄 疗法法张山山150中西中西李四李四120西西王五王五018中中刘六刘六070中中赵七七
20、135中西中西孙八八029西西哑变量化哑变量化姓名姓名性性别年年龄X1X2张山山15001李四李四12010王五王五01800刘六刘六07000赵七七13501孙八八02910Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 注意事项注意事项2.自变量的筛选 不同的筛选方法有时会产生不同的模型。实际工作中可同时采用这些方法，然后根据专业的可解释性、模型的节约性和资料采集的方便性等，决定采用何种方法的计算结果。Wednesday, Wed
21、nesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 注意事项注意事项3.交互作用 交互作用的分析十分复杂，应根据临床意义与实际情况酌情使用。Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024 注意事项注意事项4. 多分类logistic回归 心理疾病分为精神分裂症、抑郁症、神经官能症等（名义变量名义变量nomi
22、nal variables)；疗效评价分为无效、好转、显效、痊愈(有序变量有序变量ordinal variables)。 参见第17章应变量Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024SPSS软件计算sAnalyzes Regressions Binary LogisticsDependent: ysCovariates: x1 x8s Method: Forward WardsSaves Predicted Valuess Pro
23、babilitiess Group membershipsOptions CI for exp 95%s Probability for Stepwises Entry: 0.1 Removal 0.15Wednesday, Wednesday, Wednesday, Wednesday, August 21, August 21, August 21, August 21, 2024202420242024DATA samp16_1;INPUT x_ray grade stage age acid nodes;CARDS;.;PROC LOGISTIC DESCENDING;MODEL no
24、des=x_ray grade stage age acid/RISKLIMITS;OUTPUT OUT=pred PROB=pred;PROC PRINT DATA=pred;RUN; The SAS System 22:07 Monday, November 29, 2005 1 The LOGISTIC Procedure Model Information Data Set WORK.SAMP16_1 Response Variable nodes Number of Response Levels 2 Number of Observations 53 Model binary lo
25、git Optimization Technique Fishers scoring Response Profile Ordered Total Value nodes Frequency 1 1 20 2 0 33 Probability modeled is nodes=1. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 72.252 60.12
26、6 SC 74.222 71.948 -2 Log L 70.252 48.126 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr ChiSq Likelihood Ratio 22.1264 5 0.0005 Score 19.4514 5 0.0016 Wald 13.1406 5 0.0221 The SAS System 22:07 Monday, November 29, 2005 2 The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates
27、 Standard Wald Parameter DF Estimate Error Chi-Square Pr ChiSq Intercept 1 0.0618 3.4599 0.0003 0.9857 x_ray 1 2.0453 0.8072 6.4208 0.0113 grade 1 0.7614 0.7708 0.9759 0.3232 stage 1 1.5641 0.7740 4.0835 0.0433 age 1 -0.0693 0.0579 1.4320 0.2314 acid 1 0.0243 0.0132 3.4230 0.0643 The SAS System 22:0
28、7 Monday, November 29, 2005 2 The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr ChiSq Intercept 1 0.0618 3.4599 0.0003 0.9857 x_ray 1 2.0453 0.8072 6.4208 0.0113 grade 1 0.7614 0.7708 0.9759 0.3232 stage 1 1.5641 0.7740 4.0835 0.0
29、433 age 1 -0.0693 0.0579 1.4320 0.2314 acid 1 0.0243 0.0132 3.4230 0.0643 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits x_ray 7.732 1.589 37.614 grade 2.141 0.473 9.700 stage 4.778 1.048 21.783 age 0.933 0.833 1.045 acid 1.025 0.999 1.051 Association of Predicted Probabilitie
30、s and Observed Responses Percent Concordant 84.5 Somers D 0.694 Percent Discordant 15.2 Gamma 0.696 Percent Tied 0.3 Tau-a 0.332 Pairs 660 c 0.847 Wald Confidence Interval for Adjusted Odds Ratios Effect Unit Estimate 95% Confidence Limits x_ray 1.0000 7.732 1.589 37.614 grade 1.0000 2.141 0.473 9.7
31、00 stage 1.0000 4.778 1.048 21.783 age 1.0000 0.933 0.833 1.045 acid 1.0000 1.025 0.999 1.051 Obs no x_ray grade stage age acid nodes _LEVEL_ pred 1 1 0 1 1 64 40 0 1 0.25511 2 2 0 0 1 63 40 0 1 0.14633 3 3 1 0 0 65 46 0 1 0.21842 4 4 0 1 0 67 47 0 1 0.06459 。 。 。 。 50 50 1 0 1 64 89 1 1 0.80302 51 51 0 1 0 59 99 1 1 0.29880 52 52 1 1 1 68 126 1 1 0.94215 53 53 1 0 0 61 136 1 1 0.76730
